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The exact probability p(n,2) can be calculated either by using Fibonacci numbers, p(n,2) = + or by solving a direct recurrence relation leading to the same result. For higher values of k {\displaystyle k} , the constants are related to generalizations of Fibonacci numbers such as the tribonacci and tetranacci numbers.
The theory of Bayesian inference is used to derive the posterior distribution by combining the prior distribution and the likelihood function which represents the information obtained from the experiment. The probability that this particular coin is a "fair coin" can then be obtained by integrating the PDF of the posterior distribution over the ...
Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and checking which side is showing when it lands, in order to randomly choose between two alternatives. It is a form of sortition which inherently has two possible outcomes.
In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In theoretical studies, the assumption that a coin is fair is often made by referring to an ideal coin.
For example, a fair coin toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to 1 ⁄ 2. Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly 1 ⁄ 2.
The entropy of the unknown result of the next toss of the coin is maximized if the coin is fair (that is, if heads and tails both have equal probability 1/2). This is the situation of maximum uncertainty as it is most difficult to predict the outcome of the next toss; the result of each toss of the coin delivers one full bit of information.
It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and p would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and p would be the probability of tails).
Until the advent of computer simulations, Kerrich's study, published in 1946, was widely cited as evidence of the asymptotic nature of probability. It is still regarded as a classic study in empirical mathematics. 2,000 of their fair coin flip results are given by the following table, with 1 representing heads and 0 representing tails.